Chapter 8, Problem 51CR

To determine

The solution of the equation $5\left(x-3\right)+1=16$ ,

Answer to Problem 51CR

  $x=6$

Explanation of Solution

Given:

The equation, $5\left(x-3\right)+1=16$

Concept Used:

Left distributive property: $a\left(b+c\right)=a\cdot b+a\cdot c$

• To get rid of a number in addition from one side, subtract the same number from both sides of equal sign.
• To get rid of a number in subtraction from one side, add the same number both sides of equal sign.
• To get rid of a number in multiplication from one side, divide the same number from both sides of equal sign.
• To get rid of a number in division from one side, multiply the same number both sides of equal sign.

Rules of Addition/ Subtraction:

• Two numbers with similar sign always get added and the resulting number will carry the similar sign.
• Two numbers with opposite signs always get subtracted and the resulting number will carry the sign of larger number.

Rules of Multiplication/ Division:

• The product/quotient of two similar sign numbers is always positive.
• The product/quotient of two numbers with opposite signs is always negative.

Calculation:

In order to solve the given equation $5\left(x-3\right)+1=16$ for the variable, isolate the variable term x on one side by performing some basic algebraic operations to get rid of the other numbers and terms associated with it.

Here to isolate x on left side, first use Left distributive property and then subtract the like terms and then add both sides by $14$ and then divide both sides by $5$ and then simplify further as shown below,

  $\begin{array}{c}5\left(x-3\right)+1=16\\ 5\cdot x+5\cdot \left(-3\right)+1=16\\ 5x-15+1=16\\ 5x-14=16\\ 5x-14+14=16+14\\ 5x=30\\ \frac{5x}{5}=\frac{30}{5}\\ x=6\end{array}$

Thus, the solution of the given equation is $x=6$ .

Now in order to check the solution of the equation, substitute $x=6$ in the given equation and check whether the left hand side and right hand side are equal. So, it gives

  $\begin{array}{c}5\left(x-3\right)+1=16\\ 5\left(3\right)+1=16\\ 15+1=16\\ 16=16\text{True statement}\end{array}$

Thus, the solution of the given equation is $x=6$ .